Geosci. Model Dev., 10, 3411–3423, 2017 https://doi.org/10.5194/gmd-10-3411-2017 © Author(s) 2017. This work is distributed under the Creative Commons Attribution 3.0 License. Optimizing the parameterization of deep mixing and internal seiches in one-dimensional hydrodynamic models: a case study with Simstrat v1.3 Adrien Gaudard1, Robert Schwefel2, Love Råman Vinnå2, Martin Schmid1, Alfred Wüest1,2, and Damien Bouffard1,2 1Eawag, Swiss Federal Institute of Aquatic Science and Technology, Surface Waters, Research and Management, Seestrasse 79, 6047 Kastanienbaum, Switzerland 2École Polytechnique Fédérale de Lausanne, Physics of Aquatic Systems Laboratory, Margaretha Kamprad Chair, EPFL-ENAC-IIE-APHYS, 1015 Lausanne, Switzerland Correspondence to: Adrien Gaudard ([email protected]) and Damien Bouffard ([email protected]) Received: 10 October 2016 – Discussion started: 29 November 2016 Revised: 1 July 2017 – Accepted: 10 August 2017 – Published: 18 September 2017 Abstract. This paper presents an improvement of a one- environment, simultaneously reacting to external forces and dimensional lake hydrodynamic model (Simstrat) to char- acting on their surroundings. The complex hydrodynamic acterize the vertical thermal structure of deep lakes. Using processes occurring in stratified lakes are mainly governed physically based arguments, we refine the transfer of wind by the combination of surface heat flux and wind stress energy to basin-scale internal waves (BSIWs). We consider (Bouffard and Boegman, 2012). The former sets up a den- the properties of the basin, the characteristics of the wind sity stratification by warming the near-surface water, which time series and the stability of the water column to filter and floats on top of the cold deep water. This stratification pat- thereby optimize the magnitude of wind energy transferred tern isolates the lower parts of the lake (hypolimnion) from to BSIWs. We show that this filtering procedure can signif- the surface layer (epilimnion) and acts as a physical barrier icantly improve the accuracy of modelled temperatures, es- reducing vertical fluxes. The latter, wind stress, brings mo- pecially in the deep water of lakes such as Lake Geneva, for mentum into the system and thereby contributes to mixing. which the root mean square error between observed and sim- Notably, the action of wind stress on a stratified basin leads to ulated temperatures was reduced by up to 40 %. The mod- internal waves (seiches), rerouting the energy at various spa- ification, tested on four different lakes, increases model ac- tial and temporal scales (Wüest and Lorke, 2003; Wiegand curacy and contributes to a significantly better reproduction and Carmack, 1986). Basin-scale internal waves (hereafter of seasonal deep convective mixing, a fundamental parame- BSIWs) play a crucial role in the transport of mass and mo- ter for biogeochemical processes such as oxygen depletion. mentum in the lake, driving horizontal dispersion and vertical It also improves modelling over long time series for the pur- mixing, with important implications for biogeochemical pro- pose of climate change studies. cesses (Bouffard et al., 2013; Umlauf and Lemmin, 2005). In situ measurements, laboratory experiments and theoret- ical considerations have shown that the response of a strat- ified basin to wind depends on the strength, duration and 1 Introduction homogeneity of the wind field as well as the geometry of the basin and the stratification of the water column (Vale- 1.1 Hydrodynamics of vertical mixing in lakes rio et al., 2017; Valipour et al., 2015; Stevens and Imberger, 1996; Mortimer, 1974). The typical assumption consists in Lakes are recognized as sentinels of changes in climate and considering the stratified water body as a two-layer system catchment processes (Shimoda et al., 2011; Adrian et al., with different densities and thicknesses. The wave period of 2009). They have multiple intricate interactions with their Published by Copernicus Publications on behalf of the European Geosciences Union. 3412 A. Gaudard et al.: A case study with Simstrat v1.3 ble for temperature and thus density homogenization in mid- latitude water bodies. In deep lakes, however, the vertical mixing caused by con- vective and wind-induced turbulent processes (Imboden and Wüest, 1995) may be insufficient to mix the water column down to the deepest layers. In this case, the deepest part of the lake remains separated from the homogenized upper part (Jankowski et al., 2006; Straile et al., 2003). The extent of deep mixing is of critical importance for reoxygenation of the deep water, and therefore for water quality. Insufficient oxygen supply often affects lake ecosystems and can trigger the release of harmful compounds and reduced substances such as iron or manganese (Friedrich et al., 2014; Beutel and Figure 1. Period of the first longitudinal mode BSIW as a function Horne, 1999). Comprehension and accurate prediction of the of the day of year in Lake Geneva. Points are coloured based on extent of deep mixing are thus vital for proper management 2 2 the observed maximum water column stability N . Values of N < of water resources, especially in deep lakes. Moreover, deep −4 −2 10 s correspond to rare instances of complete lake overturn, mixing, as a key process of heat transfer to the hypolimnion, i.e. unstratified water columns, and are consequently not shown. partly governs its response to climate change (Ambrosetti The figure was constructed from more than 50 years of monitoring and Barbanti, 1999). However, the complex vertical pathway data at the deepest location of the lake. of kinetic energy makes modelling of the thermal structure challenging. longitudinal or transversal standing waves (TBSIW (s)) can then be estimated with the Merian formula (Bäuerle, 1994; 1.2 Numerical modelling Merian, 1828): Over the past decades, many numerical models have been −1=2 developed to allow for prediction and understanding of hy- 2 ρ2 − ρ1 h1h2 TBSIW D 2L n g ; (1) drodynamics in lakes and reservoirs. The algorithms and as- ρ2 h1 C h2 sumptions applied by these models vary greatly, as does their where L (m) is the length of the basin at the interface depth complexity, which ranges from simple box models to in- (in the direction of the excitation), g (ms−2) is the acceler- tricate three-dimensional models (Stepanenko et al., 2016; −3 ation of gravity, ρ1 and ρ2 (kgm ) are the densities of the Wang et al., 2016; Ji, 2008; Hodges et al., 2000). For climate upper and lower layers, h1 and h2 (m) are the layer thick- change studies, long-time lake simulations are needed, often nesses and n is the number of wave nodes. utilizing vertical one-dimensional (1-D) models (Wood et al., In most lakes, the zone of abrupt temperature change, 2016; Butcher et al., 2015; Komatsu et al., 2007; Hostetler commonly referred to as the thermocline or metalimnion, and Small, 1999). These models are computationally inex- progressively deepens from spring to autumn. The temper- pensive (Goyette and Perroud, 2012), permitting long-term ature difference between the two layers, determining the simulations (several decades or longer) and facilitating pa- stratification strength, decreases from mid-summer to au- rameter estimation and sensitivity analyses. tumn. The net result of both a deepening of the thermocline Common 1-D lake models have been extensively studied, and a reduced stratification strength is a strong increase in validated and compared with different sets of observational TBSIW (Eq.1). This effect is remarkably strong in meromic- data (Thiery et al., 2014; Perroud et al., 2009; Boyce et al., tic and oligomictic lakes, which can maintain stratification 1993). They have been proven to satisfactorily simulate the throughout the whole year. Lake Geneva can be categorized seasonality of surface temperature. Still, problems arise in as oligomictic with a period of the dominant first-mode lon- reproducing the vertical mixing through the thermocline and gitudinal BSIW ranging from ∼ 60 to ∼ 600 h, as shown in the evolution of the deep water temperature (Schwefel et al., Fig. 1. As expected, the wave period greatly increases from 2016; Stepanenko et al., 2010). Lake models have often been late autumn to early spring due to weakening of the stratifi- developed for specific applications and do not explicitly in- cation. clude all relevant physical processes and interactions. For Besides wind-induced BSIWs, convection is a subtle al- instance, while processes like shear-induced turbulent mix- though important vertical energetic pathway in lakes (Read ing and solar radiation are usually modelled, the effect of et al., 2012; Wüest et al., 2000). This process may be trig- internal waves is commonly not parameterized (Stepanenko gered by surface cooling or density currents (Thorpe et al., et al., 2010). As a result, the estimation of bottom tempera- 1999). The generally weak winds over perialpine lakes in- ture appears to be significantly less accurate than that of sur- crease the relative importance of convective processes in face temperature (Goyette and Perroud, 2012). This leads to these systems. Notably, cooling during winter is responsi- large discrepancies between different models under similar Geosci. Model Dev., 10, 3411–3423, 2017 www.geosci-model-dev.net/10/3411/2017/ A. Gaudard et al.: A case study with Simstrat v1.3 3413 running conditions (Stepanenko et al., 2013). Model accu- 2.1 Parameterization of internal waves racy can be improved by including BSIW parameterization and subsequent mixing (Schwefel et al., 2016). The problem Simstrat parameterizes the total BSIW energy Eseiche (J) has been partly tackled for small basins (Stepanenko et al., as a balance between production Pseiche (W) and loss 2014), but not for large, deep lakes. Lseiche (W) of seiche energy (Goudsmit et al., 2002): An inherent weakness of 1-D models is the neglect of dEseiche horizontal processes. These models may be able to repro- D P − L : (2) dt seiche seiche duce fine vertical structures, but generally perform calcula- tions that are horizontally averaged over the whole simu- Within the standard system of equations of a k–" model, lation domain (Komatsu et al., 2007).